Consider the case when no \(Z_i\)  is measured, and \(X_i\)  is a scalar. In order for \(E(X_i\mid X_{ij}^*)\)  to be linear, additive models require \(E(U_{ij}\mid X_{ij}^*)\)  to be linear, and multiplicative models require \(E\lbrace (1+\delta _lU_{il})^{-1}\mid X_{il}^*\rbrace \)  to be constant. If we have two observations from the generalized error model, \(X_{i1}^*\)  and \(X_{i2}^*\) , then direct calculations show that \(E(X_{i1}^*\mid X_{i2}^*)\)  will be linear under the same conditions that render \(E(X_i\mid X_{i2}^*)\)  linear. This applies symmetrically to \(E(X_{i2}^*\mid X_{i1}^*)\) . Checking the goodness of fit of a linear model between any two proxies in turn checks the adequacy of a linear approximation. This highlights the relationship between our methodology and instrumental variable approaches based on regressing a measurement of the truth on an instrument [1]}. These results justify both the theoretical conditions under which a linear model is warranted, and a technique for checking whether linearity holds approximately. The modified regression calibration procedure also produces asymptotically normal estimators.
